Quantum–Classical Approach for Calculations of Absorption and Fluorescence: Principles and Applications

Absorption and fluorescence spectroscopy techniques provide a wealth of information on molecular systems. The simulations of such experiments remain challenging, however, despite the efforts put into developing the underlying theory. An attractive method of simulating the behavior of molecular systems is provided by the quantum–classical theory—it enables one to keep track of the state of the bath explicitly, which is needed for accurate calculations of fluorescence spectra. Unfortunately, until now there have been relatively few works that apply quantum–classical methods for modeling spectroscopic data. In this work, we seek to provide a framework for the calculations of absorption and fluorescence lineshapes of molecular systems using the methods based on the quantum–classical Liouville equation, namely, the forward–backward trajectory solution (FBTS) and the non-Hamiltonian variant of the Poisson bracket mapping equation (PBME-nH). We perform calculations on a molecular dimer and the photosynthetic Fenna–Matthews–Olson complex. We find that in the case of absorption, the FBTS outperforms PBME-nH, consistently yielding highly accurate results. We next demonstrate that for fluorescence calculations, the method of choice is a hybrid approach, which we call PBME-nH-Jeff, that utilizes the effective coupling theory [GelzinisA.;J. Chem. Phys.2020, 152, 05110332035455] to estimate the excited state equilibrium density operator. Thus, we find that FBTS and PBME-nH-Jeff are excellent candidates for simulating, respectively, absorption and fluorescence spectra of real molecular systems.

7. Apply the FBTS formula, Eq. (8), and a similar one for PBME-nH to calculate the dipole moment operator I ; ( 1 ), ( 1 ) at each time moment in the range ∈ [ 1 , 2 ]. Note that this operator depends on the values ( 1 ), ( 1 ) parametrically. Indeed, in the quantum-classical framework,ˆI is calculated via the subsystem variables (see Eq. (8)) that, in turn, are connected to the bath variables through the Hamilton's equations.
8. According to the Monte Carlo (MC) integration scheme, calculate the th sample of the dipole-dipole correlation function (Eq. (20)) at each time moment in the range ∈ [ 1 , 2 ] as Here, we used the result for the orientational averaging, Eq. (21). Note that the Wigner image of the excited state bath equilibrium density operator, W B ( , , 1 ), from Eq. (20) is no longer present in Eq. (S1) because the equilibrium values ( 1 ), ( 1 ) calculated at each iteration of the MC loop are distributed according to the W B ( , , 1 ) function. 9. Repeat Steps 1-8 MC times to obtain MC samples of the dipole-dipole correlation function.
The algorithm for the hybrid approaches, FBTS-Jeff and PBME-nH-Jeff, is largely the same, except for the Steps 3-5. They are modified as follows: 3. Proceed to Step 4.
4. Calculate the subsystem density matrix,ˆS ( 1 ), using the effective-coupling theory. 1 (This should actually be performed only once, prior to starting the MC loop, if no disorder is present)
Our implementations of the above algorithms are available on Gitlab. 2

II. ABSORPTION LINESHAPES CALCULATED USING THE FULL-CUMULANT EXPANSION THEORY
In this section, we briefly compare the absorption lineshapes calculated using the ctR theory and the full-cumulant expansion 3 (FCE) approach. Figure S1 shows the absorption spectra of a dimer studied in the main text; three sample cases where the ctR results are not perfect are demonstrated. As we can see, the difference between the FCE and ctR results is insignificant. However, a more pronounced difference should be expected if coherence transfer takes place in the system since the possibility of such a process is totally neglected in the ctR theory. 3,4 The FCE theory, on the other hand, is capable of taking this effect into account, although this comes at an additional computational cost. 3 Figure S1. Absorption spectra of a dimer calculated using the ctR, FCE, and HEOM theories. The system parameters are taken from the default set analyzed in the main text. The parameter values specified in the figure override the corresponding default values. Figure S2 shows the fluorescence spectra of the 7 BChls FMO complex calculated using the B777 spectral density and no disorder. The FBTS and PBME-nH results were obtained using 10 8 trajectories, while the FBTS-Jeff and PBME-nH-Jeff methods already yielded converged results with just 10 6 trajectories. The obtained lineshapes are visually similar to those calculated using the Debye spectral density (shown in the main text), and similar conclusions regarding the accuracy of the quantum-classical methods can be drawn. In the low-temperature case (see Fig. S2a), the FBTS clearly leads to unphysical results since the spectrum features a region of negative intensity. The PBME-nH again predicts negative populations for some of the site-basis energy levels, hence the corresponding lineshape is likely to be incorrect as well. The PBME-nH-Jeff leads to a different curve, which has the intensities of the peaks at ∼ 12100 cm −1 and ∼ 12300 cm −1 differing by an order of magnitude. As described in the main text, this is consistent with the model of the FMO complex used in our calculations.

III. FLUORESCENCE SPECTRA OF THE FMO COMPLEX USING THE B777 SPECTRAL DENSITY
In the case of = 300 K, the effective-coupling theory may be considered correct to within at least 1% at estimating the populations, therefore, we may conclude from the lower plot in Fig. S2 that the PBME-nH estimates the equilibrium populations with a smaller error than the FBTS. The corresponding PBME-nH lineshape should thus be closer to the correct one. As in the case of Debye spectral density, the PBME-nH-Jeff results are similar to those obtained using PBME-nH. b a Figure S2. Fluorescence spectra of the 7 BChls FMO complex (upper plots) and population dynamics (lower plots) calculated using the B777 spectral density at (a) = 77 K, (b) = 300 K. The spectra are normalized to unit maximum intensity. The -coordinates of the red horizontal lines indicate the equilibrium values as given by the effective-coupling theory.
Based on the above reasoning, we conclude that the PBME-nH-Jeff should be suitable for obtaining qualitatively correct results using a reasonable amount of computational resources.